Embedding finitely presented self-similar groups into finitely presented simple groups

TL;DR

This paper proves that all finitely presented self-similar groups can be embedded into finitely presented simple groups, advancing understanding of the Boone-Higman conjecture and providing new examples involving matrix groups.

Contribution

It demonstrates that every finitely presented self-similar group embeds into a finitely presented simple group, linking to the Boone-Higman conjecture and expanding known classes of groups.

Findings

Finitely presented self-similar groups embed into finitely presented simple groups.

Certain commutator subgroups of R"over-Nekrashevych groups are finitely presented and simple.

Finitely generated subgroups of GL_n(Q) satisfy the Boone-Higman conjecture.

Abstract

We prove that every finitely presented self-similar group embeds in a finitely presented simple group. This establishes that every group embedding in a finitely presented self-similar group satisfies the Boone-Higman conjecture. The simple groups in question are certain commutator subgroups of R\"over-Nekrashevych groups, and the difficulty lies in the fact that even if a R\"over-Nekrashevych group is finitely presented, its commutator subgroup might not be. We also discuss a general example involving matrix groups over certain rings, which in particular establishes that every finitely generated subgroup of $\mathrm{GL}_n(\mathbb{Q})$ satisfies the Boone-Higman conjecture.